Regular megagon  

A regular megagon


Type  Regular polygon 
Edges and vertices  1000000 
Schläfli symbol  {1000000} t{500000} 
Coxeter diagram  
Symmetry group  Dihedral (D_{1000000}), order 2×1000000 
Internal angle (degrees)  179.99964° 
Dual polygon  self 
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
A megagon is a polygon with 1 million sides (mega, from the Greek μέγας megas, meaning "great").^{[1]}^{[2]} Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.
A regular megagon is represented by Schläfli symbol {1000000} and can be constructed as a quasiregular truncated 500000gon, t{500000}, which alternates two types of edges.
Properties
A regular megagon has an interior angle of 179.99964°.^{[1]} The area of a regular megagon with sides of length a is given by
The perimeter of a regular megagon inscribed in the unit circle is:
which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be about 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.^{[3]}
Because 1000000 = 2^{6} × 5^{6}, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular megagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a power of two, three, or six.
Philosophical application
Like René Descartes' example of the chiliagon, the millionsided polygon has been used as an illustration of a welldefined concept that cannot be visualised.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}
The megagon is also used as an illustration of the convergence of regular polygons to a circle.^{[11]}
Megagram
A megagram is an millionsided star polygon. There are 199,999 regular forms^{[12]} given by Schläfli symbols of the form {1000000/n}, where n is an integer between 2 and 500,000 that is coprime to 1,000,000. There are also 300,000 regular star figures in the remaining cases.
References
 ^ ^{a} ^{b} Darling, David J., The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. Page 249. ISBN 0471270474.
 ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, AddisonWesley, 1999. Page 505. ISBN 0201347121.
 ^ Williamson, Benjamin, An Elementary Treatise on the Differential Calculus, Longmans, Green, and Co., 1899. Page 45.
 ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
 ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0582281571.
 ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0415157927.
 ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1847063497.
 ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0198752776.
 ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
 ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0823214869.
 ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0415325056.
 ^ 199,999 = 500,000 cases  1 (convex)  100,000 (multiples of 5)  250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)
